There are some indications that the BLS has conformed to certain political biases in the past. On his website, he even points out various reliable reports where statisticians may have had meaningful political motivations to make alterations to the bundle of goods in the CPI basket for funding reasons.

He argues that politically-motivated alterations were made to lower the value of CPI when the BLS included new items in its basket of goods starting in 1998:

Quality adjustments of the hedonic, more-theoretical kind, however, have tended to reduce reported inflation meaningfully. In the article the BLS indicated that hedonic adjustments had increased prices in certain products. Those "increases," though, often were based on comparisons against prices that already had been previously adjusted and reduced based on simpler quality assessments.

There may be some reasons to believe this is the case and even some reason to believe that the government has tampered with some of its statistics.

All that being said... the very example that Williams gives as to why these hedonic adjustments are so insidious just proves how trivial and out-of-proportion the impact of these changes has been. Read his argument:

Getting more into the hedonics area, I’ll get personal. I use two personal computer systems purchased about 10 years apart for roughly the same price in nominal terms, about $800. While the most recent computer has greater memory and is faster than my old system, both systems generally perform the same tasks for me. Based on the BLS’s adjustments to computer prices, in terms of quality/hedonics, my old system should have been replaceable for about $85.00 in current dollars, which was not doable. I do have a nicer picture screen, with the new system, but I also unexpectedly had to buy a new printer, because the new system was not able to function with my antique work-horse printer. The new computer also was not able to use certain key programs that had not been rewritten to the standards of the new system. How does one compare and value such systems in the CPI? While some quality adjustment in the case of computers seems appropriate, I argue it has been heavily overdone from the practical standpoint of the average consumer.

If the only things Williams's new computer provides him with are more memory and speed, then it's hard not to believe that he didn't get ripped off. Operating software, hardware (I'd bet that 1998 computer didn't write DVDs), compatibility, functionality, heck—Wifi; obviously the '98 computer is scrap. In fact, if you look on eBay for computers built ten years ago, a Mac G4 Desktop which initially sold for $2,299 now costs $105 on eBay.

Calculations of utility are based on average consumer choices, and if Williams wanted his new computer to do exactly the same things and didn't care about new features aside from speed or memory, then he should have purchased as a $500 computer and spent the rest on a new printer.

It's still a conspiracy theory

We can clamor as much as we want about how a politician could have tried to bribe BLS officials to change the CPI system by offering the bureau money, but there is no hard evidence to prove that the BLS was motivated to change their calculations for political motives.

To boot, the data certainly don't suggest that consumers are experiencing a 10.5% inflation rate right now, no matter how difficult it is for consumers to cope with the inflation they are seeing. From economist Doug Short, who acknowledges that Williams's argument about a government bias is "thoughtful":

The more I study inflation the more convinced I am that the current BLS method of calculating inflation is reasonably sound. As a first-wave Boomer who raised a family during the double-digit inflation years of the 1970s and early 1980s, I see nothing today that is remotely like the inflation we endured at that time. Moreover, government policy, the Federal Funds Rate, interest rates in general and decades of major business decisions have been fundamentally driven by the official BLS inflation data, not the alternate CPI. For this reason I view the alternate inflation data as an interesting but ultimately useless statistical series.